Universal character of the discrete nonlinear Schrödinger equation (original) (raw)
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Universal character of the discrete nonlinear Schroedinger equation
Phys Rev a, 2007
We address the universal applicability of the discrete nonlinear Schrödinger equation. By employing an original but general top-down/bottom-up procedure based on symmetry analysis to the case of optical lattices, we derive the most widely applicable and the simplest possible model, revealing that the discrete nonlinear Schrödinger equation is "universally" fit to describe light propagation even in discrete tensorial nonlinear systems and in the presence of nonparaxial and vectorial effects.
Nonlinearity and Discreteness: Solitons in Lattices
Emerging Frontiers in Nonlinear Science, 2020
An overview is given of basic models combining discreteness in their linear parts (i.e., the models are built as dynamical lattices) and nonlinearity acting at sites of the lattices or between the sites. The considered systems include the Toda and Frenkel-Kontorova lattices (including their dissipative versions), as well as equations of the discrete nonlinear Schrödinger and Ablowitz-Ladik types, and their combination in the form of the Salerno model. The interplay of discreteness and nonlinearity gives rise to a variety of states, most important ones being discrete solitons. Basic results for 1D and 2D discrete solitons are collected in the review, including 2D solitons with embedded vorticity, and some results concerning mobility of discrete solitons.Main experimental findings are overviewed too. Models of the semi-discrete type, and basic results for solitons supported by them, are also considered, in a brief form. Perspectives for the development of topics covered the review are discussed throughout the text.
Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity
Physica D: Nonlinear Phenomena, 2009
We study the discrete nonlinear Schrödinger lattice model with the onsite nonlinearity of the general form, |u| 2σ u. We systematically verify the conditions for the existence and stability of discrete solitons in the one-dimensional version of the model predicted by means of the variational approximation (VA), and demonstrate the following: monostability of fundamental solitons (FSs) in the case of the weak nonlinearity, 2σ + 1 < 3.68; bistability, in a finite range of values of the soliton's power, for 3.68 < 2σ + 1 < 5; and the presence of a threshold (minimum norm of the FS), for 2σ + 1 ≥ 5. We also perform systematic numerical simulations to study higher-order solitons in the same general model, i.e., bound states of the FSs. While all in-phase bound states are unstable, stability regions are identified for antisymmetric double solitons and their triple counterparts. These numerical findings are supplemented by an analytical treatment of the stability problem, which allows quantitively accurate predictions for the stability features of such multipulses. When these waveforms are found to be unstable, we show, by means of direct simulations, that they self-trap into a persistent lattice breather, or relax into a stable FS, or sometimes decay completely. (J. Cuevas). universal asymptotic forms of various models based on chains of coupled oscillators. Accordingly, the solitons known in the DNLS equation represent intrinsic localized modes investigated in such chains theoretically [11] and experimentally .
Two-dimensional discrete solitons in rotating lattices
Physical Review E, 2007
We introduce a two-dimensional (2D) discrete nonlinear Schrödinger (DNLS) equation with selfattractive cubic nonlinearity in a rotating reference frame. The model applies to a Bose-Einstein condensate stirred by a rotating strong optical lattice, or light propagation in a twisted bundle of nonlinear fibers. Two species of localized states are constructed: off-axis fundamental solitons (FSs), placed at distance R from the rotation pivot, and on-axis (R = 0) vortex solitons (VSs), with vorticities S = 1 and 2. At a fixed value of rotation frequency Ω, a stability interval for the FSs is found in terms of the lattice coupling constant C, 0 < C < C cr (R), with monotonically decreasing C cr (R). VSs with S = 1 have a stability interval,C (S=1) cr Discrete dynamical systems represented by nonlinear lattices in one, two, and three dimensions constitute a class of models which are of fundamental interest by themselves, and, simultaneously, they find applications of paramount importance in various fields of physics.
Translationally invariant nonlinear Schrödinger lattices
Nonlinearity, 2006
The persistence of stationary and travelling single-humped localized solutions in the spatial discretizations of the nonlinear Schrödinger (NLS) equation is addressed. The discrete NLS equation with the most general cubic polynomial function is considered. Constraints on the nonlinear function are found from the condition that the second-order difference equation for stationary solutions can be reduced to the first-order difference map. The discrete NLS equation with such an exceptional nonlinear function is shown to have a conserved momentum but admits no standard Hamiltonian structure. It is proved that the reduction to the first-order difference map gives a sufficient condition for existence of translationally invariant single-humped stationary solutions. Another constraint on the nonlinear function is found from the condition that the differential advance-delay equation for travelling solutions admits a reduction to an integrable normal form given by a third-order differential equation. This reduction gives a necessary condition for existence of single-humped travelling solutions. The nonlinear function which admits both reductions defines a fourparameter family of discrete NLS equations which generalizes the integrable Ablowitz-Ladik lattice. Particular travelling solutions of this family of discrete NLS equations are written explicitly.
Theory of Nonlinear Matter Waves in Optical Lattices
Modern Physics Letters B, 2004
We consider several effects of the matter wave dynamics which can be observed in Bose-Einstein condensates embedded into optical lattices. For low-density condensates we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross-Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force, and lattice defects affect the nonlinear matter waves. *
Communications in Nonlinear Science and Numerical Simulation, 2020
We study coupled unstaggered-staggered soliton pairs emergent from a system of two coupled discrete nonlinear Schrödinger (DNLS) equations with the self-attractive on-site self-phasemodulation nonlinearity, coupled by the repulsive cross-phase-modulation interaction, on 1D and 2D lattice domains. These mixed modes are of a "symbiotic" type, as each component in isolation may only carry ordinary unstaggered solitons. While most work on DNLS systems addressed symmetric on-site-centered fundamental solitons, these models give rise to a variety of other excited states, which may also be stable. The simplest among them are antisymmetric states in the form of discrete twisted solitons, which have no counterparts in the continuum limit. In the extension to 2D lattice domains, a natural counterpart of the twisted states are vortical solitons. We first introduce a variational approximation (VA) for the solitons, and then correct it numerically to construct exact stationary solutions, which are then used as initial conditions for simulations to check if the stationary states persist under time evolution. Twocomponent solutions obtained include (i) 1D fundamental-twisted and twisted-twisted soliton pairs, (ii) 2D fundamental-fundamental soliton pairs, and (iii) 2D vortical-vortical soliton pairs. We also highlight a variety of other transient dynamical regimes, such as breathers and amplitude death. The findings apply to modeling binary Bose-Einstein condensates, loaded in a deep lattice potential, with identical or different atomic masses of the two components, and arrays of bimodal optical waveguides.
Introduction to Solitons in Photonic Lattices
Springer Series in Optical Sciences, 2009
We present a review on wave propagation in nonlinear photonic lattices: arrays of optical waveguides made of nonlinear media. Such periodic structures provide an excellent environment for the direct experimental observations and theoretical studies of universal phenomena arising from the interplay between nonlinearity and Bloch periodicity. In particular, we review one-dimensional and two-dimensional lattice solitons, spatial gap solitons, and vortex lattice solitons.
Physical Review A, 2010
We investigate solitons and nonlinear Bloch waves in Bose-Einstein condensates trapped in optical lattices. By introducing specially designed localized profiles of the spatial modulation of the attractive nonlinearity, we construct an infinite number of exact soliton solutions in terms of the Mathieu and elliptic functions, with the chemical potential belonging to the semi-infinite bandgap of the optical-lattice-induced spectrum. Starting from the exact solutions, we employ the relaxation method to construct generic families of soliton solutions in a numerical form. The stability of the solitons is investigated through the computation of the eigenvalues for small perturbations, and also by direct simulations. Finally, we demonstrate a virtually exact (in the numerical sense) composition relation between nonlinear Bloch waves and solitons.